For a family of expanded degenerations \(\pi :\mathtt {X}_{S}\rightarrow S\)  over \((\xi , S)\)  and an \(S\) -flat coherent sheaf \(\mathcal {F}\)  on \(\mathtt {X}_{S}\) , one has the Hilbert homomorphism \(P_{\mathcal {F}_{s}}^{\mathcal {E}_{\mathcal {X}_{\mathbf {m}(\xi (s))}}}: K^0(\mathcal {X}_{\mathbf {m}(\xi (s))})\rightarrow \mathbb {Z}\) 
on the fiber over any point \(s\in S\)  where \(\xi : S\rightarrow \mathbb {A}^{k+1}\)  is a \(\mathbb {A}^1\) -map and \(\mathbf {m}:\mathbb {A}^{k+1}\rightarrow \mathbb {A}^1\)  is the multiplication map. When \(\mathbf {m}(\xi (s))=0\) , \(\mathcal {F}_{s}\)  is defined on the fiber over \(s\)  which is isomorphic to \(\mathcal {X}_{0}[l]\) , where \((l+1)\)  is equal to the number of zeros in the coordinate of \(\xi (s)\)  by Proposition REF . Since \(S\)  is connected, the similar argument in the proof of [[1]}, Lemma 4.3] shows that there is a group homomorphism \(P:K^0(\mathcal {X})\rightarrow \mathbb {Z}\)  such that for any \(s\in S\)  we have \(P_{\mathcal {F}_{s}}^{\mathcal {E}_{\mathcal {X}_{\mathbf {m}(\xi (s))}}}=P\) .
With the categorical interpretation of the stack \(\mathfrak {PT}_{\mathfrak {X}/\mathfrak {C}}\) , for a fixed group homomorphism \(P:K^{0}(\mathcal {X})\rightarrow \mathbb {Z}\) , we define \(\mathfrak {PT}_{\mathfrak {X}/\mathfrak {C}}^{P}\)  to be the subfunctor of \(\mathfrak {PT}_{\mathfrak {X}/\mathfrak {C}}\)  parameterizing stable orbifold PT pairs with fixed Hilbert homomorphism \(P\) . Then \(\mathfrak {PT}_{\mathfrak {X}/\mathfrak {C}}^{P}\)  is an open and closed subfunctor of \(\mathfrak {PT}_{\mathfrak {X}/\mathfrak {C}}\)  (it is empty unless the associated generalized Hilbert polynomial is of degree one). By
Theorem REF , \(\mathfrak {PT}_{\mathfrak {X}/\mathfrak {C}}^{P}\)  is a Deligne-Mumford stack locally of finite type over \(\mathbb {A}^1\) .
